SOLUTION: if sn the sum of first n terms of an AP is given by Sn=3nsq-4n then find nth term

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Question 955505: if sn the sum of first n terms of an AP is given by Sn=3nsq-4n then find nth term
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
S%5Bn%5D , the sum of first n terms of an AP (Arithmetic progression or Arithmetic sequence) , is given by
S%5Bn%5D=%282a%5B1%5D%2B%28n-1%29d%29%2An%2F2 , where
a%5B1%5D is the first term, and
d is the common difference.

The problem says that S%5Bn%5D=3n%5E2-4n (or so I guess),
and that is true for all and any natural number values of n .
So, we have to solve the equation
%282a%5B1%5D%2B%28n-1%29d%29%2An%2F2=3n%5E2-4n ,
which is true for all and any natural number values of n .
%282a%5B1%5D%2B%28n-1%29d%29%2An%2F2=3n%5E2-4n<-->%282a%5B1%5Dn%2Bdn%5E2-dn%29%2F2=3n%5E2-4n<-->%28dn%5E2%2B2a%5B1%5Dn-dn%29%2F2=3n%5E2-4n<-->%28d%2F2%29n%5E2%2B%28%282a%5B1%5D-d%29%2F2%29n=3n%5E2-4n
If that must be true for all and any natural number values of n ,
then the coefficients of the terms in n%5E2 and n must be the same, so
system%28d%2F2=3%2C%282a%5B1%5D-d%29%2F2=-4%29-->system%28d=6%2C%282a%5B1%5D-6%29%2F2=-4%29-->system%28d=6%2Ca%5B1%5D-3=-4%29-->system%28d=6%2Ca%5B1%5D=-1%29 .
For any AP, the nth term is given by
a%5Bn%5D=a%5B1%5D%2B%28n-1%29d , where
a%5B1%5D is the first term, and
d is the common difference.
With system%28d=6%2Ca%5B1%5D=-1%29 , a%5Bn%5D=-1%2B6%28n-1%29<-->a%5Bn%5D=6n-7